At most locations, there are no cheapest dates to rent, in absolute terms. The cheapest dates to rent are often (but not always) a week or two out (if the facility is not close to sold out by then, of course). So I just book "pay later" rates, and find that I can almost always lower my rate by amending my reservation a week or two out (though it depends on the location, some locations are expensive no matter when you book).

So for what you want to do, what you would really need is not a flexible date calendar (because that only tells what's cheapest at that moment, which may be close to useless), but rather historical data for that location to see patterns (like annual major conventions or other events) that raise prices on certain dates each year.

Some sites such as Orbitz used to show you a multi-day grid (which would show you if Saturday or Sunday was cheaper at your specified pickup-time and your specified drop-off time), but Orbitz doesn't seem to have that any more, and in case you had to specify pick up and return time and it only showed something like +/- 3 days. I don't know if any site does that much any more.